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The Superposition Principle & Standing Waves

Instructor: David Wood

David has taught Honors Physics, AP Physics, IB Physics and general science courses. He has a Masters in Education, and a Bachelors in Physics.

After completing this lesson, you will be able to explain what the superposition principle is, and how this leads to the formation of standing waves. A short quiz will follow.

What is the Superposition Principle?

One of the first mathematical facts we learn as small children is that 1 + 1 = 2. We learn how to add. But pure numbers aren't the only thing that we can add together. If you pour water into a bucket that already contains some water, the water level gets higher. And if you take a beam of light, and add a second beam of light on top of the first, you will get an even brighter beam. This second case, where you put two things directly over the top of each other, is called superposition.

In physics, we want to understand how the universe works. And so we need to understand what happens when you superimpose two things. And the superposition principle is part of how we explain what happens. The superposition principle says that if you superimpose two or more stimuli, the result you get will be the sum of what you would have got from each of them.

Or in other words, if you put two light beams with brightnesses A and B on top of each other, the total brightness will be A+B.

The superposition principle holds for almost all everyday situations involving waves, like light and sound waves. However, in some complicated physics circumstances, for example laser beams, the principle does not apply.

Wave Interference

Did you know that you can add waves, too? We've already talked about light, and light is a wave. But you can add sound waves. You can even add waves on a slinky or spring. And in many cases, waves add together according to the superposition principle.

A Wave on a Slinky
A Wave on a Slinky

Waves contain peaks and troughs, and when you superimpose two waves on top of each other, those peaks and troughs add together. Maybe this would make more sense if we went through an example of how you could do that.

Let's say you and a friend are playing with a slinky. You hold the slinky at each end, and one of you moves your arm up and down to send a wave along it. The wave goes down to the other end, bounces off your friend's hand and comes back the other way, until it eventually dies out.

But what happens if you both start a wave at the same time? You send a wave from your end, and your friend sends a wave from his end. What happens when they collide in the middle?

It turns out they pass through each other. But while they're passing through, the two waves are superimposed. When two peaks are superimposed, they add up, and you get a bigger peak. When two troughs are superimposed, they add up, and you get a bigger trough. And when a peak and a trough are superimposed, they cancel out and you get no wave at all.

Combinations of Peaks and Troughs
Combinations of Peaks and Troughs

It's kind of like adding. A peak and a peak is like adding 1 + 1 = 2. A trough and a trough is like adding -1 + -1 = -2. And a trough and a peak is like adding -1 + 1 = 0.

Standing Waves

If you and your friend keep sending waves down the slinky over and over again, and move your arm up and down fast enough, you can get something called a standing wave.

This is where two waves going in opposite directions interfere with each other in a consistent way. It creates a pattern that looks something like this:

The Movement of a Standing Wave Over Time
Standing Wave

A standing wave is a vibration of a system where some points remain fixed, while others vibrate with the maximum amplitude. It looks like parts of the slinky are just moving up and down. But there really ARE two waves. They're just interfering in ways that create this interesting pattern.

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